Finite Group Databasebeta

← #29712438

Subgroups — #29712438

Order 15616.

1424 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840451C211yesmaximal, characteristic, complemented
2C2×#16839913C211yesmaximal, characteristic, complemented
3#21449169C211yesmaximal, characteristic
4#21446963C212yesmaximal, complemented
5#21446823C212yesmaximal, complemented
6#21446252C211yesmaximal, characteristic, complemented
7#21446171C212yesmaximal, complemented
8#21446133C212yesmaximal, complemented
9#21446116C211yesmaximal, characteristic, complemented
10#21445914C211yesmaximal, characteristic, complemented
11C2×C61×#418C211yesmaximal, characteristic, complemented
12C22×#11758200C2212yes—
13#16840451—24no—
14#16840451—24no—
15C2×#14519770—21no—
16C2×#14519769—21no—
17C2×#14519768C2211yescharacteristic, complemented
18C2×#14519759—21no—
19C2×#14519757C2211yescharacteristic, complemented
20C2×#14519756C2211yescharacteristic
21C2×#14519748C2212yes—
22C2×#14519746—21no—
23C2×#14519742C2211yescharacteristic, complemented
24C2×C4×C22⋊D61C2211yescharacteristic
25C2×#14519734—21no—
26C2×C4⋊C4×D61C2211yescharacteristic, complemented
27#16839913—24no—
28C2×#14519702C2212yescomplemented
29C2×#14519698C2212yes—
30C2×C22⋊#92418C2211yescharacteristic
31C2×#14519689C2212yescomplemented
32#16839676—21no—
33#16839670—21no—
34#16839665C2212yes—
35#16839622C2211yescharacteristic
36#16839618—21no—
37#16839525C2212yescomplemented
38#16839524C2212yescomplemented
39#16839524C2212yescomplemented
40#16839516C2212yescomplemented
41C23×D4×C61C2211yescharacteristic
42C22×M16×C61C2211yescharacteristic
43C2×#117×C61C2211yescharacteristic
44C61×#418—24no—
45C2×#102×C61C2212yes—
46C2×#102×C61C2212yes—
47C22×Dic122C2311yescharacteristic
48C22×C22⋊D61—22no—
49C22×C22⋊D61D411yescharacteristic
50C22×C4×D61—21no—
51C22×C4×D61C2311yescharacteristic
52C2×#11758200—24no—
53C2×#11758200—24no—
54C2×#11758200—42no—
55C22×#92418C2312yes—
56#14519770—28no—
57#14519769—28no—
58#14519768—22no—
59#14519768—22no—
60#14519759—28no—
61#14519757—22no—
62#14519757—22no—
63#14519756—22no—
64#14519756D414yescomplemented
65#14519748—24no—
66#14519748—42no—
67#14519746—44no—
68#14519742—22no—
69#14519742—22no—
70C4×C22⋊D61—22no—
71C4×C22⋊D61—22no—
72#14519734—28no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61C2312yes—
76C2×C4⋊Dic61C2311yescharacteristic
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196D411yescharacteristic
82C4⋊C4×D61—22no—
83C4⋊C4×D61—22no—
84C2×C22⋊Dic61—21no—
85C2×C22⋊Dic61—21no—
86C2×C22⋊Dic61—22no—
87C2×C22⋊Dic61—22no—
88C2×C22⋊Dic61—22no—
89C2×C22⋊Dic61D411yescharacteristic
90C2×#11758198—21no—
91C2×#11758198D411yescharacteristic
92C2×C4×Dic61—21no—
93C2×C4×Dic61C2311yescharacteristic
94#14519702—42no—
95#14519698—24no—
96C22⋊#92418—22no—
97#14519689—42no—
98#14519684—21no—
99C25×C61—21no—
100C22×D4×C61—24no—
101C22×D4×C61—22no—
102C22×D4×C61—22no—
103C22×D4×C61—21no—
104C22×D4×C61—42no—
105C22×D4×C61C2312yes—
106C23×C4×C61C2311yescharacteristic
107C2×M16×C61—22no—
108C2×M16×C61—42no—
109C22×C8×C61C2312yes—
110#117×C61—22no—
111#117×C61D414yescomplemented
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C4⋊C4×C61C2311yescharacteristic
114C2×C22⋊C4×C61—21no—
115C2×C42×C61—21no—
116#102×C61—24no—
117#102×C61—24no—
118C24×C61—22no—
119C24×C61—44no—
120C24×C61—42no—
121C24×C61—42no—
122C24×C61—42no—
123C24×C61—82no—
124C24×C61C2×D411yescharacteristic
125C23×D61C2×D411yescharacteristic
126C2×D4×C61—28no—
127C2×D4×C61—22no—
128C2×D4×C61—22no—
129C2×D4×C61—22no—
130C2×D4×C61—22no—
131C2×D4×C61—22no—
132C2×D4×C61—48no—
133C2×D4×C61—44no—
134C2×D4×C61—44no—
135C2×D4×C61—42no—
136C2×D4×C61#4114yes—
137C22×C4×C61—22no—
138C22×C4×C61—22no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D412yes—
143C22×C4×C61C2×D412yes—
144C22×C4×C61C2411yescharacteristic
145C2×C22⋊D61—22no—
146C2×C22⋊D61—22no—
147C2×C22⋊D61—41no—
148C2×C22⋊D61—44no—
149C2×C22⋊D61—44no—
150C2×C22⋊D61—82no—
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—22no—
154C22×Dic61C2×D411yescharacteristic
155C22×Dic61C2×D411yescharacteristic
156C22×Dic61C2×D411yescharacteristic
157C2×C4×D61—24no—
158C2×C4×D61—21no—
159C2×C4×D61—21no—
160C2×C4×D61—22no—
161C2×C4×D61—22no—
162C2×C4×D61—42no—
163C2×C4×D61—42no—
164C2×Dic122—22no—
165C2×Dic122—22no—
166C2×Dic122—21no—
167C2×Dic122C2×D412yes—
168M16×C61—44no—
169C2×C8×C61—22no—
170C2×C8×C61—22no—
171C2×C8×C61—22no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61C2×D412yes—
175C4⋊C4×C61C2×D412yes—
176C22⋊C4×C61—24no—
177C42×C61—24no—
178C22⋊Dic61—24no—
179C22⋊Dic61—22no—
180C22⋊Dic61—44no—
181C22⋊Dic61—44no—
182C22⋊Dic61—42no—
183C22⋊Dic61—82no—
184#11758200—48no—
185#11758198—24no—
186#11758198—22no—
187C4⋊Dic61—24no—
188C4⋊Dic61—24no—
189C4⋊Dic61—21no—
190C4⋊Dic61—22no—
191C4⋊Dic61—22no—
192C4⋊Dic61C2×D412yes—
193#11758196—24no—
194#11758196—24no—
195#11758196—24no—
196#11758196—24no—
197#11758196—22no—
198C4×Dic61—24no—
199C4×Dic61—21no—
200C4×Dic61—21no—
201C2×#92418—22no—
202C2×#92418—22no—
203C2×#92418—22no—
204C23×C61—22no—
205C23×C61—21no—
206C23×C61—21no—
207C23×C61—22no—
208C23×C61—24no—
209C23×C61—48no—
210C23×C61—42no—
211C23×C61—42no—
212C23×C61—42no—
213C23×C61—42no—
214C23×C61—42no—
215C23×C61—42no—
216C23×C61—84no—
217C23×C61—84no—
218C23×C61—88no—
219C23×C61—88no—
220C23×C61#12012yes—
221C23×C61C22×D411yescharacteristic
222C22×D61—21no—
223C22×D61—22no—
224C22×D61—21no—
225C22×D61—22no—
226C22×D61AGL(1,ℤ/8)12yes—
227D4×C61—24no—
228D4×C61—24no—
229D4×C61—48no—
230D4×C61—42no—
231D4×C61—42no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—22no—
235C2×C4×C61—24no—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—42no—
241C2×C4×C61#12012yes—
242C2×C4×C61C22×D411yescharacteristic
243C2×C4×C61C22×D411yescharacteristic
244C2×C4×C61C2×#4112yes—
245C2×C4×C61H(5,𝔽2)12yes—
246C22⋊D61—44no—
247C22⋊D61—88no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—22no—
251C2×Dic61—21no—
252C2×Dic61—22no—
253C2×Dic61—22no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—22no—
257C2×Dic61—42no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—42no—
261C2×Dic61—42no—
262C2×Dic61—42no—
263C4×D61—48no—
264C4×D61—41no—
265C4×D61—41no—
266C4×D61—42no—
267Dic122—24no—
268Dic122—41no—
269Dic122—41no—
270C8×C61—42no—
271#92418—42no—
272#23982—611nomaximal, Sylow
273C22×C61—22no—
274C22×C61—22no—
275C22×C61—22no—
276C22×C61—42no—
277C22×C61—42no—
278C22×C61—44no—
279C22×C61—44no—
280C22×C61—44no—
281C22×C61—816no—
282C22×C61—82no—
283C22×C61—82no—
284C22×C61—82no—
285C22×C61—82no—
286C22×C61C2×#12011yescharacteristic
287C22×C61#54612yes—
288C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
289C22×C61#57511yescharacteristic
290C22×C61#57712yes—
291C2×D61—22no—
292C2×D61—24no—
293C2×D61—41no—
294C2×D61—41no—
295C2×D61—42no—
296C4×C61—22no—
297C4×C61—42no—
298C4×C61—42no—
299C4×C61#52511yescharacteristic
300C4×C61#53511yescharacteristic
301C4×C61#54612yes—
302Dic61—42no—
303Dic61—41no—
304Dic61—42no—
305Dic61—41no—
306Dic61—41no—
307Dic61—82no—
308C2×#552—611no—
309C2×#450—611no—
310C2×#418D6111yescharacteristic, complemented
311#2408—611no—
312#2025—612no—
313#2000—612no—
314#1894—611no—
315#1880—612no—
316#1874—612no—
317#1870—611no—
318#1825—611no—
319C2×C61—42no—
320C2×C61—84no—
321C2×C61#240411yescharacteristic
322C2×C61#300611yescharacteristic
323C2×C61#301011yescharacteristic
324C2×C61#331012yes—
325C2×C61#332312yes—
326D61—42no—
327C23×D4C2×D6111yescharacteristic
328C22×SD16—612no—
329C22×M16C2×D6111yescharacteristic
330#552—1224no—
331#552—1224no—
332C2×#126—1221no—
333C2×#125—611no—
334C2×#125—1221no—
335C2×#123—1221no—
336C2×C22⋊Q8—611no—
337C2×C22⋊Q8—611no—
338C2×#121—612no—
339C2×#120—1221no—
340C2×C4×Q8—611no—
341C2×C4×D4—611no—
342C2×#117—1221no—
343C2×#117C2×D6111yescharacteristic
344C22×C4⋊C4—611no—
345#450—1224no—
346#418—24no—
347C2×#103—612no—
348C2×#102—612no—
349C2×#102C2×D6112yes—
350C2×#102C2×D6112yes—
351C2×C22⋊C8—611no—
352C2×C22⋊C8—612no—
353#394—1221no—
354#392—1221no—
355#390—612no—
356#380—611no—
357#379—1221no—
358#340—612no—
359#340—612no—
360#340—612no—
361#336—612no—
362C61#2398211yescharacteristic, complemented, Sylow
363C25—21no—
364C22×Q8—611no—
365C22×D4—24no—
366C22×D4—22no—
367C22×D4—21no—
368C22×D4—22no—
369C22×D4—42no—
370C22×D4—611no—
371C22×D4—1222no—
372C22×D4C22×D6112yes—
373C23×C4—611no—
374C23×C4—1221no—
375C23×C4C22×D6111yescharacteristic
376C2×SD16—1224no—
377C2×SD16—1224no—
378C2×SD16—2442no—
379C2×M16—22no—
380C2×M16—42no—
381C22×C8—612no—
382C22×C8C22×D6112yes—
383#126—1228no—
384#125—1228no—
385#125—1222no—
386#125—1222no—
387#123—1228no—
388C22⋊Q8—614no—
389C22⋊Q8—1222no—
390C22⋊Q8—1222no—
391C22⋊Q8—1222no—
392#121—1224no—
393#121—2442no—
394#120—2444no—
395C4×Q8—1222no—
396C4×Q8—1222no—
397C4×D4—1222no—
398C4×D4—1222no—
399#117—22no—
400#117—1228no—
401#117C22⋊D6114yescomplemented
402C2×C4⋊C4—611no—
403C2×C4⋊C4—611no—
404C2×C4⋊C4—612no—
405C2×C4⋊C4—1222no—
406C2×C4⋊C4—1221no—
407C2×C4⋊C4—1221no—
408C2×C4⋊C4—1221no—
409C2×C4⋊C4—1221no—
410C2×C4⋊C4—1221no—
411C2×C4⋊C4—1221no—
412C2×C4⋊C4—1222no—
413C2×C4⋊C4C22×D6111yescharacteristic
414C2×C4⋊C4C22×D6111yescharacteristic
415C2×C22⋊C4—21no—
416C2×C22⋊C4—611no—
417C2×C22⋊C4—611no—
418C2×C22⋊C4—1221no—
419C2×C22⋊C4—1221no—
420C2×C22⋊C4—1222no—
421C2×C22⋊C4—1222no—
422C2×C22⋊C4—1222no—
423C2×C22⋊C4—1221no—
424C2×C42—21no—
425C2×C42—611no—
426C2×C42—1221no—
427#103—2442no—
428#102—24no—
429#102—24no—
430#102—1224no—
431C22⋊C8—1222no—
432C22⋊C8—2442no—
433#95—1221no—
434C24—22no—
435C24—44no—
436C24—42no—
437C24—42no—
438C24—42no—
439C24—82no—
440C24—611no—
441C24D4×D6111yescharacteristic
442C2×Q8—612no—
443C2×Q8—1221no—
444C2×Q8—1222no—
445C2×Q8—1222no—
446C2×D4—28no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—48no—
453C2×D4—44no—
454C2×D4—44no—
455C2×D4—42no—
456C2×D4—1222no—
457C2×D4—1222no—
458C2×D4—4882no—
459C2×D4—2441no—
460C2×D4—2444no—
461C2×D4—2444no—
462C2×D4#1175822214yes—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—611no—
468C22×C4—611no—
469C22×C4—611no—
470C22×C4—1224no—
471C22×C4—1221no—
472C22×C4—1221no—
473C22×C4—1221no—
474C22×C4—1221no—
475C22×C4—1222no—
476C22×C4—1222no—
477C22×C4—1222no—
478C22×C4—2442no—
479C22×C4—2442no—
480C22×C4D4×D6112yes—
481C22×C4D4×D6111yescharacteristic
482C22×C4C2×C22⋊D6112yes—
483C22×C4C23×D6111yescharacteristic
484SD16—2448no—
485M16—44no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—1222no—
490C2×C8—1222no—
491C2×C8—1222no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—612no—
495C4⋊C4—1224no—
496C4⋊C4—1224no—
497C4⋊C4—1224no—
498C4⋊C4—1224no—
499C4⋊C4—1224no—
500C4⋊C4—1221no—
501C4⋊C4—1222no—
502C4⋊C4—1224no—
503C4⋊C4—1222no—
504C4⋊C4—1222no—
505C4⋊C4C2×C22⋊D6112yes—
506C4⋊C4C2×C22⋊D6112yes—
507C22⋊C4—24no—
508C22⋊C4—1224no—
509C22⋊C4—1222no—
510C22⋊C4—1224no—
511C22⋊C4—1222no—
512C22⋊C4—4882no—
513C22⋊C4—2444no—
514C22⋊C4—2444no—
515C22⋊C4—2442no—
516C42—24no—
517C42—1224no—
518C42—1221no—
519C42—1221no—
520C23—21no—
521C23—21no—
522C23—22no—
523C23—22no—
524C23—24no—
525C23—48no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—84no—
533C23—84no—
534C23—88no—
535C23—88no—
536C23—612no—
537C23—1221no—
538C23—1221no—
539C23—1222no—
540C23—1222no—
541C23#1451974512yes—
542C23C2×D4×D6111yescharacteristic
543Q8—1224no—
544Q8—2441no—
545Q8—2441no—
546D4—24no—
547D4—24no—
548D4—48no—
549D4—42no—
550D4—42no—
551D4—4888no—
552D4—2444no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—1221no—
563C2×C4—1222no—
564C2×C4—1221no—
565C2×C4—1222no—
566C2×C4—1221no—
567C2×C4—1221no—
568C2×C4—1221no—
569C2×C4—1222no—
570C2×C4—1222no—
571C2×C4—2448no—
572C2×C4—2442no—
573C2×C4—2442no—
574C2×C4—2442no—
575C2×C4—2441no—
576C2×C4—2441no—
577C2×C4—2442no—
578C2×C4—2441no—
579C2×C4—2441no—
580C2×C4—2442no—
581C2×C4#1451974512yes—
582C2×C4C2×D4×D6111yescharacteristic
583C2×C4C22×C22⋊D6111yescharacteristic
584C2×C4C2×#1175822212yes—
585C2×C4#1451982212yes—
586C8—42no—
587C8—2442no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—42no—
592C22—42no—
593C22—44no—
594C22—44no—
595C22—44no—
596C22—816no—
597C22—82no—
598C22—82no—
599C22—82no—
600C22—82no—
601C22—1224no—
602C22—1222no—
603C22—2441no—
604C22—2442no—
605C22—2441no—
606C22C2×#1451974511yescharacteristic
607C22#1684040212yes—
608C22AGL(1,ℤ/8)×D6111yescharacteristic
609C22#1684055811yescharacteristic
610C22#1684057212yes—
611C4—22no—
612C4—42no—
613C4—42no—
614C4—4882no—
615C4—2441no—
616C4—2441no—
617C4—2442no—
618C4—2441no—
619C4—2442no—
620C4#1684027311yescharacteristic
621C4#1684032911yescharacteristic
622C4#1684040212yes—
623C2—42no—
624C2—84no—
625C2—2442no—
626C2#2144913811yescharacteristic
627C2#2145506711yescharacteristic
628C2#2145510411yescharacteristic
629C2#2145784112yes—
630C2#2145803112yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#418C2
socleC23×C61C22×D4
prime radical#21449169C2
hypercenterC2×#418D61
hypocenterC61#23982
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#418

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#16840451C2semidirect
2. C2×#16839913C2semidirect
4. #21446963C2semidirect
5. #21446823C2semidirect
6. #21446252C2semidirect
7. #21446171C2semidirect
8. #21446133C2semidirect
9. #21446116C2semidirect
10. #21445914C2semidirect
11. C2×C61×#418C2semidirect
17. C2×#14519768C22semidirect
19. C2×#14519757C22semidirect
23. C2×#14519742C22semidirect
26. C2×C4⋊C4×D61C22semidirect
28. C2×#14519702C22semidirect
31. C2×#14519689C22semidirect
37. #16839525C22semidirect
38. #16839524C22semidirect
39. #16839524C22semidirect
40. #16839516C22semidirect
64. #14519756D4semidirect
111. #117×C61D4semidirect
310. C2×#418D61semidirect
362. C61#23982semidirect
401. #117C22⋊D61semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168404512
C2×#168399132
#214491692
#214469632
#214468232
#214462522
#214461712
#214461332
#214461162
#214459142
C2×C61×#4182
#2398261